Zariskian Filtrations: 2 (K-Monographs in Mathematics) - Tapa blanda

In Commutative Algebra certain /-adic filtrations of Noetherian rings, i.e. the so-called Zariski rings, are at the basis of singularity theory. Apart from that it is mainly in the context of Homological Algebra that filtered rings and the associated graded rings are being studied not in the least because of the importance of double complexes and their spectral sequences. Where non-commutative algebra is concerned, applications of the theory of filtrations were mainly restricted to the study of enveloping algebras of Lie algebras and, more extensively even, to the study of rings of differential operators. It is clear that the operation of completion at a filtration has an algebraic genotype but a topological fenotype and it is exactly the symbiosis of Algebra and Topology that works so well in the commutative case, e.g. ideles and adeles in number theory or the theory of local fields, Puisseux series etc, .... . In Non­ commutative algebra the bridge between Algebra and Analysis is much more narrow and it seems that many analytic techniques of the non-commutative kind are still to be developed. Nevertheless there is the magnificent example of the analytic theory of rings of differential operators and 1J-modules a la Kashiwara-Shapira.

This book is the first to present a complete theory of filtrations on associative rings, combining techniques stemming from number theory related to valuations, with facts originating in the study of rings of differential operators on varieties. It deals with the homological algebra part of the theory via an innovative use of graded ring theory applied to the Rees ring of a filtration. This leads to a completely new approach to extensions of valuations, regularity conditions on noncommutative algebras, and geometric aspects of rings of differential operators, and provides new applications related to deformations of algebras, gauge algebras and other physics-related objects.
Audience: This volume will be of interest to graduate students and researchers in different fields of mathematics and mathematical physics.